What is the thickness of the atmosphere?

In summary, the conversation discusses different ways to estimate the thickness of Earth's atmosphere, with the main focus being on using assumptions of thermal and diffusive equilibrium to calculate the difference in chemical potential between the external and internal potential. This approach leads to an estimate of 10^5m, which is off by a factor of 6 due to an underestimate of the mass of air molecules. Other methods, such as calculating the kinetic energy of gas molecules or the ratio of pressure difference to pressure needed to balance gravity, also result in a similar estimate of around 9 km. There is also a mention of the atmosphere potentially ending at the same point where particles in intersolar space are in equilibrium with those escaping Earth.
  • #1
Shing
144
1
I was thinking of a Fermi-question: the thickness of atmosphere with diffusive equilibrium. And I estimated roughly 10^{5}m (where it should be ~10^{4}m). The difference of order of magnitude to real thickness is 1 (from Wiki).

I had a lot of fun, and I am looking for interesting ways other than chemical potential to estimate the thickness of Earth's atmosphere.

---

Basically, I started with assumptions:

- thermal equilibrium
- diffusive equilibrium

such that I can set up the difference in chemical potential is zero, that is equal to the difference in external chemical potential (gravitational potential) plus the difference in internal potential (assuming air molecular just ideal gas); mathematically:

$$\Delta\mu=0=\Delta\mu_{ext}+\Delta\mu_{int}$$

which is

$$0=\Delta\mu_{gra}+\Delta\mu_{indeal}$$

$$0=-GM_{earth}m\Big( \frac{1}{R+\Delta h} - \frac{1}{R} \Big)+k_BT\ln\Big(\frac{n_2}{n_1}\Big)$$

Setting the ratio of n_2, n_1 to be 0.1 (so I defined the height where density of air goes from unity - ground - to 0.1 as "thickness"), and m the mass of air molecular to be 10^{-26}kg, T to be 300K, then we will eventually obtain: $$\Delta h \approx 10^5m$$
 
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  • #2
You underestimate the mass of a molecule in the air by a factor 6, that leads to most of the difference.
There are many ways to get the same expression, but they all lead to good estimates.
 
  • #3
mfb said:
You underestimate the mass of a molecule in the air by a factor 6
Thanks for telling me!

mfb said:
There are many ways to get the same expression, but they all lead to good estimates.
I am all ears :)
 
  • #4
You can calculate the kinetic energy of a gas molecule, and see how high it can get if it moves straight up with this energy.
You can calculate the ratio of pressure difference to pressure needed to balance a pocket of gas against gravity.
And probably a few more.

They all boil down to ##h=\frac{kT}{mg} \approx 9\,km## with additional prefactors of ln(2) or similar depending on what you do.
 
  • #5
I would tend to think that the atmosphere ends in the same way the heliosphere ends - i.e., where the particles in intersolar space are in equilibrium with the particles escaping Earth. Of course, Von Karman had a great definition in which a standard aircraft body would need to be going so fast to maintain altitude that it would be in orbit.
 

Related to What is the thickness of the atmosphere?

1. What is the average thickness of the atmosphere?

The average thickness of the atmosphere is approximately 100 kilometers (62 miles) from the Earth's surface.

2. How does the thickness of the atmosphere vary at different altitudes?

The thickness of the atmosphere decreases as altitude increases. At sea level, the atmosphere is the thickest, while at the top of Mount Everest, the thinnest known layer of the atmosphere exists.

3. Does the thickness of the atmosphere vary around the world?

Yes, the thickness of the atmosphere can vary depending on the location and weather conditions. For example, the atmosphere is usually thinner in areas with higher elevations and thicker in areas with lower elevations.

4. How does the thickness of the atmosphere affect climate?

The thickness of the atmosphere plays a crucial role in regulating the Earth's climate. The thicker the atmosphere, the more heat it can trap, leading to warmer temperatures. On the other hand, a thinner atmosphere can result in cooler temperatures.

5. Can the thickness of the atmosphere change over time?

Yes, the thickness of the atmosphere can change over time due to various factors such as human activities, natural phenomena, and climate change. For example, air pollution can alter the composition and thickness of the atmosphere, leading to changes in climate patterns.

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